I'm not an expert, but my understanding is those brackets are just used to limit vibrations in areas where there are fast acting valves (like near a washing machine), or where the pipe runs through studs/joists and vibration can occur. Just use your discretion. If you think the pipe has a chance of vibrating and causing unwanted noise, use the insulator.

Thanks, gents. So I scattered 'em around to a handful of locations where I thought made sense, and will continue with that approach. Still can't make sense out of the what/where/why my plumber did before me, but oh well.

The screw holes seem superfluous since I have to pry these with a screwdriver to remove them, but since they're there, I guess I can burn a few more pennies driving anchors.

So in summary, it sounds like the concensus is to use some sort of pipe insulator on ALL lines (not just fast-acting values) and to place either every stud/joist or every other. Yes?

Now to go slightly off topic... For floor penetrations, Uponor has a 1/2" metal straight-through support. Is that to be used on ALL lines as well? What exactly does this protect against? (It being metal it seems it would not be for movement as I presume it would chafe.) Is this installed on top of the sole/bottom plate of a wall or where?